# Minkowski’s Loop Fractal Antenna Dedicated to Sixth Generation (6G) Communication

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Fractal Antennas, Fractals Geometry, Minkowski Fractal Characteristics

#### 2.1. About the Fractal Antennas

- Compact/short dimension;
- Reduced perimeter;
- Multi-band frequency;
- Conformal typology.

- Antenna gain losing;
- Complicated composition;
- Smaller benefits in dimension according to early recurrence.

#### The 6G Radio Frequency

#### 2.2. About the Fractal Character: Minkowski’s Loop

_{n−1}into five subsections r

_{n}− c

_{n}− b

_{n}− c

_{n}− r

_{n}and repeating it over and over again. Between the values of the five segments is the following constitutive relation a

_{n−1}= 2r

_{n}+ b

_{n}, in which c

_{n}is the fractal deep of the generator in the n-th Minkowski recurrent relation (we start from a straight line, and we obtain the step generator, height/indentation equal to r

_{n}), Figure 1, baseline. Another indicator used, named ${\delta}_{n}$, is the iteration factor in the respective iteration and is noted as ${\delta}_{n}={C}_{n}/{B}_{n}$. Both values of equality, respectively C

_{n}and ${\delta}_{n}$ at the n-th iteration, are adaptable and will be optimized based on the design performances of the developed antenna.

^{D}, a power type law, as in the calculus formula for D

_{1}and a

_{2}. In this situation, the fractal dimension, D, can be achieved, as in the solution of the next equation:

_{1}= L

_{1}/Lo, respectively, a

_{2}= L

_{2}/Lo, Figure 1b.

_{1}= x (on ox axis) and a

_{2}= y (on oy axis) functioning as independent variables is presented. The value of D is greater than 1 and less than 2 (in our chart less than 1.99, to be exact).

## 3. Minkowski Fractal Antenna

- Length = 0.03 m; Width = 0.028 m; StripLineWidth = 0.0008 m; SlotLength = 0.004 m;
- SlotWidth = 0.00585 m; Height = 0.001 m; GroundPlaneLength = 0.05 m;
- GroundPlaneWidth = 0.03 m; FractalCenterOffset (m) = [0 0]; Tilt (deg) = 0; TiltAxis = [1 0 0].

_{r}(air) =1 and which is present in a layer of 0.00004 m. The absolute dielectric permittivity of the classical vacuum is ε

_{0}= 8.8541 × 10

^{−12}F⋅m

^{−1}.

#### The Revised Minkowski Geometry Figure of Fractal

## 4. Results and Discussion

**E**and

**H**, but with a higher density in the area of the two geographical poles. In the left corner of the sketch is the Minkowski flat fractal antenna, designed for the fourth iteration.

_{11}and return loss graphics are presented.

_{11}), and respectively, the return loss for (b), of the Minkowski type fractal antennas for two, three and four iterations.

_{11}). These mistaking overlaps do occur, however, from the fact that these quantities defined above all describe the reflection of a wave propagating from a reference pack, either that it is a terminal transmission line or that it is a grid of preset circuits, ultimately.

#### Fractal Antenna Measurements

_{r}= 4.4. The fractal antenna is fed from the normal position by coaxial cable having the inner and outer diameters of the SMA connector. The scale factor of antennas is 1/3 and the stage of iteration is n = 3. The size of the substrate and the patch are the same. At the end of the modeling, the Gerber files are generated, useful for printing the antenna wiring on the dielectric material (the PCB design has to be completed with a specialized device to strictly observe the fractal dimensions). In the measurements effectuated with the fractal antenna obtained, the VDI - Erickson Power Meters (PM5B) were used. This power meter, covering both analog and digital carriers, is a calibrated calorimeter-style power meter for 75 GHz to >3 THz applications. It offers power measurement ranges from 1 µW up to 200 mW. The PM5B is the de facto standard for frequency > 100 GHz power measurement and can be used in measurements such as VSWR for antenna and cable, antenna return loss and cable return loss to measure forward power and measure reflected power.

## 5. Horn Antenna versus Minkowski Fractal Antenna

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A drawing with the four iterations made (

**a**–

**d**), starting from the Minkowski initiator. (

**a**) the initial start (square ring); (

**b**) the 1st iteration; (

**c**) the 2nd iteration; (

**d**) the initial start (square ring) plus the first three iterations.

**Figure 2.**The 3D graphical representation of the fractal dimension D, depending on the variables a

_{1}and a

_{2}.

**Figure 16.**Minkowski fractal antenna for 2 iterations: (

**a**) Self-reflection coefficient S

_{11}, (

**b**) Return loss.

**Figure 17.**Minkowski fractal antenna for 3 iterations: (

**a**) Self-reflection coefficient S

_{11}, (

**b**) Return loss.

**Figure 18.**Minkowski fractal antenna for 4 iterations: (

**a**) Self-reflection coefficient S

_{11}, (

**b**) Return loss.

Band | Frequency | WR-Size |
---|---|---|

D | 110 GHz to 170 GHz | WR-6 |

G | 140 GHz to 220 GHz | WR-5 |

G | 170 GHz to 260 GHz | WR-4 |

G | 220 GHz to 325 GHz | WR-3 |

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**MDPI and ACS Style**

Paun, M.-A.; Nichita, M.-V.; Paun, V.-A.; Paun, V.-P.
Minkowski’s Loop Fractal Antenna Dedicated to Sixth Generation (6G) Communication. *Fractal Fract.* **2022**, *6*, 402.
https://doi.org/10.3390/fractalfract6070402

**AMA Style**

Paun M-A, Nichita M-V, Paun V-A, Paun V-P.
Minkowski’s Loop Fractal Antenna Dedicated to Sixth Generation (6G) Communication. *Fractal and Fractional*. 2022; 6(7):402.
https://doi.org/10.3390/fractalfract6070402

**Chicago/Turabian Style**

Paun, Maria-Alexandra, Mihai-Virgil Nichita, Vladimir-Alexandru Paun, and Viorel-Puiu Paun.
2022. "Minkowski’s Loop Fractal Antenna Dedicated to Sixth Generation (6G) Communication" *Fractal and Fractional* 6, no. 7: 402.
https://doi.org/10.3390/fractalfract6070402